# If you combine 360.0 mL of water at 25.00°C and 120.0 mL of water at 95.00°C, what is the final temperature of the mixture?

The theory behind this is that the heat absorbed by the water sample at room temperature and the heat lost by the hot water sample will be equal.

This is a negative sign because heat loss is represented by a minus sign.

Here, your go-to formula will be

Thus, the two samples' respective temperature changes will be

You'll get this

Four sign figs are used to round the result.

You can find information about redoing the calculations using the actual densities of water at those two temperatures here. As a practice, you should try it.

Water Density (https://tutor.hix.ai)

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To find the final temperature, you can use the principle of conservation of energy:

(Q_{\text{lost}} = Q_{\text{gain}})

Where (Q_{\text{lost}}) is the heat lost by the hot water, and (Q_{\text{gain}}) is the heat gained by the cold water.

The heat lost or gained can be calculated using the formula:

(Q = mc\Delta T)

Where (Q) is the heat lost or gained, (m) is the mass of the substance, (c) is the specific heat capacity, and (\Delta T) is the change in temperature.

Since the specific heat capacity of water is 4.18 J/(g°C), we can use the formula to find the final temperature.

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To find the final temperature of the mixture, we can use the principle of conservation of energy, which states that the total energy of an isolated system remains constant. We can use the equation ( q = mcΔT ), where ( q ) represents the heat absorbed or released, ( m ) represents the mass of the substance, ( c ) represents the specific heat capacity, and ( ΔT ) represents the change in temperature.

First, we need to calculate the heat gained or lost by each sample of water. Using the formula ( q = mcΔT ), we can find the heat for each sample separately.

For the first sample (water at 25.00°C): ( q_1 = m_1cΔT_1 )

For the second sample (water at 95.00°C): ( q_2 = m_2cΔT_2 )

Next, we can use the fact that the total heat gained by the cooler water equals the total heat lost by the hotter water to find the final temperature of the mixture. The equation is: ( q_1 + q_2 = 0 )

Finally, we rearrange the equation to solve for the final temperature (( T_f )): ( T_f = \frac{m_1cΔT_1 + m_2cΔT_2}{m_1c + m_2c} )

Plugging in the given values and constants: ( T_f = \frac{(360.0 g)(4.18 J/g°C)(T_f - 25.00°C) + (120.0 g)(4.18 J/g°C)(95.00°C - T_f)}{(360.0 g)(4.18 J/g°C) + (120.0 g)(4.18 J/g°C)} )

After solving this equation, we find the final temperature of the mixture to be ( T_f = 41.4°C ).

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